Czechoslovak Mathematical Journal, Vol. 62, No. 4, pp. 889-900, 2012

On co-ordinated quasi-convex functions

M. Emin Özdemir, Ahmet Ocak Akdemir, Çetin Yıldız

M. E. Özdemir, Atatürk University, Erzurum, Turkey, e-mail: emos@atauni.edu.tr; A. O. Akdemir, Ağrı İbrahim Çeçen University, Ağrı, Turkey, e-mail: ahmetakdemir@agri.edu.tr; Ç. Yıldız, Atatürk University, Erzurum, Turkey, e-mail: yildizc@atauni.edu.tr

Abstract: A function $f I\rightarrow\mathbb{R}$, where $I\subseteq\mathbb{R}$ is an interval, is said to be a convex function on $I$ if
f( tx+( 1-t) y) \leq tf( x) +(1-t) f( y)
holds for all $x,y\in I$ and $t\in[ 0,1]$. There are several papers in the literature which discuss properties of convexity and contain integral inequalities. Furthermore, new classes of convex functions have been introduced in order to generalize the results and to obtain new estimations. We define some new classes of convex functions that we name quasi-convex, Jensen-convex, Wright-convex, Jensen-quasi-convex and Wright-quasi-convex functions on the co-ordinates. We also prove some inequalities of Hadamard-type as Dragomir's results in Theorem 5, but now for Jensen-quasi-convex and Wright-quasi-convex functions. Finally, we give some inclusions which clarify the relationship between these new classes of functions.

Keywords: co-ordinate, quasi-convex, Wright-quasi-convex, Jensen-quasi-convex

Classification (MSC 2010): 26D15

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